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Which of the following series converge, and which diverge?

  • $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{2}+1}$

  • $\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^{n}}$

  • $\displaystyle\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^{2}+n}}$

MY ATTEMPT

The first one converges due to the comparison test. Indeed, one has \begin{align*} \sum_{n=1}^{\infty}\frac{1}{n^{2}+1} \leq \sum_{n=1}^{\infty}\frac{1}{n^{2}} \end{align*} where the last series is the $p$-series with $p = 2 > 1$.

The second series does converge due to the ratio test. Indeed, one has \begin{align*} \lim_{n\to\infty}\frac{(n+1)!}{(n+1)^{n+1}}\times\frac{n^{n}}{n!} = \lim_{n\to\infty}\left(\frac{n}{n+1}\right)^{n} = \lim_{n\to\infty}\left(1 - \frac{1}{n+1}\right)^{n} = e^{-1} < 1 \end{align*}

Finally, for the third series it suffices to notice that $n^{2} + n \leq 2n^{2}$. Thence we conclude that it diverges. Indeed, one has \begin{align*} \sum_{n=1}^{\infty}\frac{1}{\sqrt{n^{2}+n}} \geq \frac{1}{\sqrt{2}}\sum_{n=1}^{\infty}\frac{1}{n} \longrightarrow +\infty \end{align*}

Is the wording of my solutions good? Any comments and contributions are appreciated.

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  • $\begingroup$ Seems legit to me. $\endgroup$ – FearfulSymmetry Jul 1 '20 at 2:05
  • $\begingroup$ Yep, looks good to me. $\endgroup$ – K.defaoite Jul 1 '20 at 3:04
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Most of this is correct and well-worded. Where you have used the phrase "harmonic series with $p = 2$", you really should say "$p$-series with $p = 2$. The harmonic series is $\sum_{n=1}^\infty \frac{1}{n}$, whereas $p$-series are series of the form $\sum_{n=1}^\infty \frac{1}{n^p}$.

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  • $\begingroup$ I would have called the comparison series used in the first one a "p-series". I reserve "harmonic series" for the special case of $p=1$. $\endgroup$ – Jason DeVito Jul 1 '20 at 2:40
  • $\begingroup$ @JasonDeVito Good call; I missed that. Why not edit in the correction? It is community wiki after all. $\endgroup$ – user804886 Jul 1 '20 at 2:41
  • $\begingroup$ @JasonDeVito thanks for the comment. I haved edited my answer in order to include your suggestion. $\endgroup$ – user1337 Jul 1 '20 at 3:22
  • $\begingroup$ @BrickByBrick The so-called p-series (dumb name) is also called the hyperharmonic series. $\endgroup$ – bof Jul 1 '20 at 3:31
  • $\begingroup$ I understand that StackExchange encourages editting of others people's work, but I just don't really like to do it, especially when it comes to changing the content (as opposed to simple grammar/spelling edits.) $\endgroup$ – Jason DeVito Jul 1 '20 at 4:55

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